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--- math104-s22:notes:lecture_15 [2022/03/07 22:45]
pzhou created
+++ math104-s22:notes:lecture_15 [2022/03/09 22:37] (current)
pzhou [Connectedness]
@@ Line 26: / Line 26: @@
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+. Compactness is an absolute (or intrinsic) property of a metric space. If  $X$  is a metric space, and  $K \In X$  is a subset, when we say  $K$  is compact, we mean  $K$  as a 'stand-alone' metric space (totally forgetting about  $X$ , but only using the distance function inherited from  $X$ ) is compact.
+. We proved last time: If  $K \In X$  is (open cover) compact, then  $K$  is closed and bounded. (Discussion: if you replace open cover compact by sequential compactness, can you prove the two conclusions directly (without using the equivalence of the two definitions)?)
+. We know that  $[0,1]$  is sequentially compact (by Heine-Borel theorem), can you show that  $[0,1]^2$  is sequentially compact? (Hint: given a sequence  $(p_n)$  in  $[0,1]^2$ , first show that you can pass to subsequence to make the sequence of the first coordinate converge, then ...)

Lecture Notes